Homogeneity and the risk-neutral density

(a) Show that the Black–Scholes call price satisfies $c(\lambda S,\lambda K)=\lambda c(S,K)$ for $\lambda>0$, and deduce from Euler's theorem that $c=S\,\partial_Sc+K\,\partial_Kc$. (b) Use (a) and $\Delta_c=e^{-qT}N(d_1)$ to show $\partial_Kc=-e^{-rT}N(d_2)$. (c) Show, for any model in which $c(K)=e^{-rT}\mathbb E^{\mathbb Q}[(S_T-K)^+]$ and $S_T$ has a density $f$ under $\mathbb Q$, that $\partial_Kc=-e^{-rT}\mathbb Q(S_T>K)$ and $\partial_K^2c=e^{-rT}f(K)$ (Breeden–Litzenberger). What does (c) imply about the sign of a butterfly price?